IntroductionIn blackjack, players are sometimes offered the opportunity to make one or more side bets related to the hands they may be dealt. This bet operates independently of the main blackjet bet. One such bet relates to perfect pairs. A perfect pair occurs when a player is dealt two identical cards - for example, two tens of hearts. In deciding whether to place such a bet, a player should weigh the winning payout odds against the probability of getting a perfect pair.
What is the chance of getting a perfect pair in a six-deck shoe (52*6=312 cards)?
Simple SolutionGiven the first card is whatever it is, let's say it's a ten of hearts, there are 5 remaining such cards in the six-deck shoe, out of 311 remaining cards in the shoe. The probability of a perfect pair is therefore:
Alternate SolutionThe alternate solution is a bit more involved. Of all the combinations of hands possible from a six-deck shoe, how many combinations are perfect pairs?
The number of possible combinations is the number of ways to choose 2 items from 312 items:
To get the number of perfect pair combinations, it's helpful to start with one example, so again let's say the ten of hearts. There are 6 tens of hearts in the shoe. The number of combinations involving two tens of hearts is the number of ways to choose 2 items from 6 items:
We then multiply this number by 52 to get the number of perfect pair combinations for all 52 unique cards. We end up therefore with the following equation:
The alternate, more complex solution, gives the same answer as the simple solution above.
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